How can I create a diffuse map that looks and reacts like a Vertex Weight map?
You have two options:
Image sequence

With the canvas object selected, set the format to image, then set the UV map and output path/format:


Bake the image sequence, then load it into your material with an image node:



Result:

Vertex paint:
You can also bake to vertex colors and use a node setup like this, using an attribute node to get the vertex color: